Pick's Theorem

Okay, good, now let's do it another way. There's a method of
calculating the area of any polygon on a geoboard quickly and easily
using what's called Pick's theorem. It's almost too good to be true,
but Pick's theorem computes the area of an arbitrary polygon simply by
counting pegs. Note that the polygon in Figure 7 touches ten pegs and surrounds
six pegs. Now if we take the number of boundary pegs (which is 10 in
this case) and divide it in half, add the number of interior pegs
(which is 6) and subtract one, we get

which should agree with what you obtained earlier by summing the areas
of the polygon's constituent squares and triangles.

Let's check to see that Pick's theorem holds for two of the simple
examples in Figure 6. The rectangle in
Figure 6a, for instance, has ten boundary pegs and two interior pegs.
By Pick's theorem, the area is

which checks with our earlier calculation via the area formula
A = b x h. Similarly, for the parallelogram in
Figure 6b, we have

We leave it to you to use Pick's theorem to verify the areas of the
triangle and trapezoid in Figures 6c and 6d.

Pick's theorem, then, can be stated as follows. Let b be the
number of boundary pegs and i be the number of interior pegs of
any polygon on the geoboard. Then the area of the polygon is given
by

With a little practice, you'll be able to simply look at a polygon and
compute its area in your head.